DOCSLIB.ORG

Permutation Matrices Whose Convex Combinations Are Orthostochastic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Permutation Matrices Whose Convex Combinations Are Orthostochastic View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Permutation Matrices Whose Convex Combinations Are Orthostochastic Yik-Hoi Au-Yeung and Che-Man Cheng Department of Mathematics University of Hong Kong Hong Kong Submitted by Richard A. Brualdi ABSTRACT Let P,, ., t’,,, be n x n permutation matrices. In this note, we give a simple necessary condition for all convex combinations of P,, . , Pm to be orthostochastic. We show that for n < 15 the condition is also sufficient, but for n > 15 whether the condition is suffkient or not is still open. 1. INTRODUCTION An n X n doubly stochastic (d.s.) matrix (aij> is said to be orthostochastic (o.s.) if there exists an n X n unitary matrix (a,,> such that aij = IuijJ2. OS. matrices play an important role in the consideration of generalized numerical ranges (for example, see [2]). For some other properties of OS. matrices see [4], [5], and [6]. For n > 3, it is known that there are d.s. matrices which are not 0.s. (for example, see [4]), and for n = 3 there are necessary and sufficient conditions for a d.s. matrix to be O.S. [2], but for n > 3 no such conditions have been obtained. We shall denote by II,, the group of all n x n permuta- tion matrices and by @n the set of all rr X n OS. matrices. Let P,, . , P,, E II,. We shall give a necessary condition for conv( P,, . , P,,,} c On, where “conv” means “convex hull of,” and then show that the same condition is also sufficient for n < I5. 2. THE CONDITION The following concept will play an important role. DEFINITION 1. Let P = (pij) and Q = (qij) be in H,. Then we say that P and Q are cmnpkmentary if, for any I < i, j, h, k < n, pij = phk = qik = 1 implies qhj = I (consequently, qij = qhk = pik = I implies phj = I). LINEARALGEBRAANDZTSAPPLlCATIONS150:243-253 (1991) 243 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas,New York, NY 10010 0024-3795/91/$3.50 244 YIK-HOI AU-YEUNG AND CHE-MAN CHENG The following two lemmas are trivial. LEMMA 1. Let P, Q, R, S E II,. Then P and Q are complementary zyand only ay RPS and RQS are complementary. LEMMA 2. Let R, S E II,. Then M E q, ayand only afRMS E en. We shall denote the n X n identity matrix by I,,. LEMMA 3. L.etP,QEII, and O<t<l. Then tP+(l-t)QEO” ifand only if P and Q are complementary. Proof. *: Let P = (pii> and Q = (yjj), and let 1 < i,j, h, k ,< n be such that pjj = phk = qik = 1. We may assume i + h and j z k; otherwise, as PEII,, we have i=h and j=k and qhj=qik =l. Suppose that qhjfl. Then qhj = 0. By considering the ith and hth rows of tP +(l - t>Q, we see that tP + (I- t)Q ~5 q,. =: From Lemmas 1 and 2, we may assume P = I,,. Since I, and Q are complementary, Q is symmetric. Hence there is S E ff,, such that S[tZ,+(l-t)Q]S-‘=J,@ ... @Jk@Zn_2k, where J~=(,!, ‘L”) for i=l,...,k and n - 2k 2 0. Consequently, tl, + (1 - t)Q E 0,. n The following theorem follows immediately from Lemma 3. THEOREM 1. Let P,, . , P,,~rI,.Zfconv(P, ,..., P,n}~@a,thenP, ,..., P, are pairwise complementary. We are now going to consider whether the condition in Theorem 1 is sufficient. Suppose that P,, . , P,,L are pair-wise complementary and one of the Pi (i = 1,. , m> is I,. Then P,, . , P,, are symmetric. For symmetric permutation matrices, we have LEMMA 3. Let P, Q E II, be symmetric. Then P and Q are complemen- tary if and only if PQ is symmetric (i.e. PQ = QP). PERMUTATION MATRICES 245 Proof. Since P, Q are symmetric and Rt = R- ’ for any R E ff,, where t means transpose, we have by Lemma 1 P and Q are complementary t) 1, PQ are complementary * PQ is symmetric 0 PQ=QP. n We now introduce another concept. DEFINITION 2. Let X c IIn and p be a prime number. Then X is called a p-set if Pp = I, for all P E X, and a p-set is called a commute p-set if PQ=QP forall P,QEX. From Lemma 1, Lemma 4, and Definition 2, we have COROLLARY 1. Let P,, ., P,,, E lI,, be symmetric and pairtiise comple- mentary. Then (PI,. , P,,,} is a commute e-set. COROLI,AKY2. Let P,, P,, . , P,,, E n,, be pairwise complementary. Then {I,, PF’P,, . , PI’P,,,} is a commute g-set. We shall first consider the structure of a commute p-set. Let G be a subgroup of S,,, the symmetric group of degree n. Then G acts on the set N = (1,. , n] in the usual way. For any x E N, we shall denote by G(x) the orbit of x under G [i.e. G(x)= (v(x):u E G}]. For any finite set X, we shall denote by 1x1 the number of elements in X. Noting that the orbits form a partition of N, we easily obtain the following lemma. LEMMA 5. Let G be a subgroup of S,, r E S,,, and x,x1. y, y1 E N such that x1 E G(x), y1 E G(y), T(x~) = yl, and TQ = UT for all v E G. Then (i) r(G(x)) = G(y) and consequently IG(x)l = IG(y)l; (ii) the restriction 7jcC,) of T on G(x) is uniquely determined by x, and yl. Let u E S,. Define P, = (pij)i,j=l,,,,,n, where pjj = aiaCjj and aik is the Kronecker delta. Then P, E n,, and the correspondence from u to P, is an 246 YIK-HOI AU-YEUNG AND CHE-MAN CHENG isomorphism from S, onto II,. If x E N and 9 C II,, then we define P,(r) = o(x) and 9(x) = {P(X): P E 9). For any two square matrices A and B, not necessarily of the same size, we denote by A@ B and A@ B the direct sum and tensor product respectively. COROLLARY 3. Let 9 be a subgroup of n,, such that &(;$,+1) ={ ~EJi+l,~~&i+,,., ;&+n*} for s=l,...,Z, where ni, i = 1,. ,I, are positive integers, and Cl = ,ni = n (hence P E 9 * P= P,@ .- . @P,, where Pi E II,C for i = 1, . , 1). Suppose that Q = (Qsl)s,b=l ,..,, l E II,, where Qst is an n,s X n, matrix such that QP = PQ for all P E 9. Then, for any l< s, t < 1, ifQst + 0, then n, = n,, Q,s, E II,%, and Qsr is uniquely determined by any of its nonzero entries. Proof. From the definition we see that if P E Kl, and x E N, then P(x) is the position of the (unique) 1 that appears in the x th column. Suppose the (a,P)th element of Q,, is 1, where 1~ LYQ n,s and 1 < j3 < n,. Then from our assumption we have s-1 s-l Cn,+aES En,+1 , i=l i i=l I t-1 t-1 Cnj+/3E8 Cnj+l , j=l i j=l I and Our result then follows from Lemma 5. n Let p be a prime number. We give two examples of commute p-sets. PERMUTATION MATRICES 247 EXAMPLE 1. Let (T be the cyclic permutation (l,~, p - 1,. ,2) ( E S,) and C,, = P, ( E n,). For any 1 >/ 1 we define the following matrices in ff,f: P/“= z 631 c3 . @.c I’ I’ /I Then the set is a commute p-set in ll),,. The subgroup #trl generated by XLII is also a commute p-set which is given by d[,]= {Cp3q~ . ~CI~:l~i,~pforj=l,...,I}. Furthermore, for any 1~ i, j < p’, there is a unique P E gtlI such that the (i, j)th element of P is 1. Consequently gt;rI(l) = (1,. , ~‘1, and, by Gd- lary 3, for any Q E fI,,/, if Q commutes with each element in XIII, then Q E &[;I]. We shall define .9t;ol = i(1)). EXAMPLE 2. Suppose that li >, 0 for i = 1,. , y and n = x7= 1~‘1. Then the subgroup (of fI,,) defined by ={PEII,:P= P,@ ... @PC,, PiELgr[l,l for i=l,..., q) 9 [I,.. ./,,I is a commute p-set. Furthermore, for l<k <y. THEOREM 2. Suppose that 9 is a subgroup of n,, und is a commute p-set. Then there exist li B 0 f&r i = 1,. , q and Q E n, such that n = Cp= ,p’l and 9 is a subgroup of the group Qdra,,,,,,,, ,,,,Q-‘. 248 YIK-HOI AU-YEUNG AND CHE-MAN CHENG Proof. We shall prove the theorem by induction on n. It is obviously true when n = 1. Assume that n > 1 and the theorem holds for all k < n. Suppose that 9 is a subgroup of II, and is a commute p-set. We may assume 191 > 1. Take P E 3 \(I,). Since Pp = I,, there exists Q E II, such that If n - mp > 0, then by considering the subgroup generated by Q- ’ PQ and using Corollary 3, we see that for any R E 9, Q-‘RQ = R,@R,, where R I E Il,,,,, and R, E II,,_-,,,,,.
Load more

Recommended publications
  • Lecture Notes: Qubit Representations and Rotations
    Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1.
    [Show full text]
  • The Geometry of Convex Cones Associated with the Lyapunov Inequality and the Common Lyapunov Function Problem∗
    Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 12, pp. 42-63, March 2005 ELA www.math.technion.ac.il/iic/ela THE GEOMETRY OF CONVEX CONES ASSOCIATED WITH THE LYAPUNOV INEQUALITY AND THE COMMON LYAPUNOV FUNCTION PROBLEM∗ OLIVER MASON† AND ROBERT SHORTEN‡ Abstract. In this paper, the structure of several convex cones that arise in the studyof Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence. Key words. Lyapunov functions and stability, Convex cones, Matrix equations. AMS subject classifications. 37B25, 47L07, 39B42. 1. Introduction and motivation. Recently, there has been considerable inter- est across the mathematics, computer science, and control engineering communities in the analysis and design of so-called hybrid dynamical systems [7,15,19,22,23]. Roughly speaking, a hybrid system is one whose behaviour can be described math- ematically by combining classical differential/difference equations with some logic based switching mechanism or rule. These systems arise in a wide variety of engineer- ing applications with examples occurring in the aircraft, automotive and communi- cations industries. In spite of the attention that hybrid systems have received in the recent past, important aspects of their behaviour are not yet completely understood.
    [Show full text]
  • Theory of Angular Momentum and Spin
    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
    [Show full text]
  • Arxiv:1909.13402V1 [Math.CA] 30 Sep 2019 Routh-Hurwitz Array [14], Argument Principle [23] and So On
    ON GENERALIZATION OF CLASSICAL HURWITZ STABILITY CRITERIA FOR MATRIX POLYNOMIALS XUZHOU ZHAN AND ALEXANDER DYACHENKO Abstract. In this paper, we associate a class of Hurwitz matrix polynomi- als with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion. Keywords: Hurwitz stability, matrix polynomials, total positivity, Markov parameters, Hankel matrices, Stieltjes positive definite sequences, quasiminors 1. Introduction Consider a high-order differential system (n) (n−1) A0y (t) + A1y (t) + ··· + Any(t) = u(t); where A0;:::;An are complex matrices, y(t) is the output vector and u(t) denotes the control input vector. The asymptotic stability of such a system is determined by the Hurwitz stability of its characteristic matrix polynomial n n−1 F (z) = A0z + A1z + ··· + An; or to say, by that all roots of det F (z) lie in the open left half-plane <z < 0. Many algebraic techniques are developed for testing the Hurwitz stability of matrix polynomials, which allow to avoid computing the determinant and zeros: LMI approach [20, 21, 27, 28], the Anderson-Jury Bezoutian [29, 30], matrix Cauchy indices [6], lossless positive real property [4], block Hurwitz matrix [25], extended arXiv:1909.13402v1 [math.CA] 30 Sep 2019 Routh-Hurwitz array [14], argument principle [23] and so on.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • Unipotent Flows and Applications
    Clay Mathematics Proceedings Volume 10, 2010 Unipotent Flows and Applications Alex Eskin 1. General introduction 1.1. Values of indefinite quadratic forms at integral points. The Op- penheim Conjecture. Let X Q(x1; : : : ; xn) = aijxixj 1≤i≤j≤n be a quadratic form in n variables. We always assume that Q is indefinite so that (so that there exists p with 1 ≤ p < n so that after a linear change of variables, Q can be expresses as: Xp Xn ∗ 2 − 2 Qp(y1; : : : ; yn) = yi yi i=1 i=p+1 We should think of the coefficients aij of Q as real numbers (not necessarily rational or integer). One can still ask what will happen if one substitutes integers for the xi. It is easy to see that if Q is a multiple of a form with rational coefficients, then the set of values Q(Zn) is a discrete subset of R. Much deeper is the following conjecture: Conjecture 1.1 (Oppenheim, 1929). Suppose Q is not proportional to a ra- tional form and n ≥ 5. Then Q(Zn) is dense in the real line. This conjecture was extended by Davenport to n ≥ 3. Theorem 1.2 (Margulis, 1986). The Oppenheim Conjecture is true as long as n ≥ 3. Thus, if n ≥ 3 and Q is not proportional to a rational form, then Q(Zn) is dense in R. This theorem is a triumph of ergodic theory. Before Margulis, the Oppenheim Conjecture was attacked by analytic number theory methods. (In particular it was known for n ≥ 21, and for diagonal forms with n ≥ 5).
    [Show full text]
  • Non-Hurwitz Classical Groups
    London Mathematical Society ISSN 1461–1570 NON-HURWITZ CLASSICAL GROUPS R. VINCENT and A.E. ZALESSKI Abstract In previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra 28 (2000) 5383–5404] it is shown that cer- tain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the spe- cial linear and special unitary groups in dimensions 8,9,11,13. We also show that all groups Sp(n, q) are not Hurwitz for q even and n =6, 8, 12, 16. In the range 11 <n<32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3), P Ω±(8, 3k), P Ω±(10,q), Ω(11, 3k), Ω±(14, 3k), Ω±(16, 7k), Ω(n, 7k) for n =9, 11, 13, PSp(8, 7k) are not Hurwitz. 1. Introduction A finite group H = 1 is called Hurwitz if it is generated by two elements X, Y satisfying the conditions X2 = Y 3 =(XY )7 = 1. A long-standing problem is that of classifying simple Hurwitz groups. The problem has been solved for alternating groups by Conder [4], and for sporadic groups by several authors with the latest result by Wilson [27]. It remains open for groups of Lie type and for classical groups. 3 2 Quite a lot is known. Groups D4(q) for (q, 3)=1, G2(q),G2(q) are Hurwitz with 3 k few exceptions, groups D4(3 ) are not Hurwitz; see Jones [10] and Malle [15, 16].
    [Show full text]
  • Analysis and Control of Linear Time–Varying Systems Exercise 2.3
    Analysis and control of linear time–varying 2 systems Before considering the actual subject matter, i.e., analysis and control design of nonlinear systems, at first linear time-varying systems or LTV systems of the form x˙ A(t)x B(t)u, t t , x(t ) x (2.1a) Æ Å È 0 0 Æ 0 y C(t)x D(t)u, t t (2.1b) Æ Å ¸ 0 with x(t) Rn, u(t) Rm, y(t) Rp and 2 2 2 n n n m p n p m A(t): RÅ R £ , B(t): RÅ R £ , C(t): RÅ R £ , D(t): RÅ R £ t 0 ! t 0 ! t 0 ! t 0 ! are considered. Here RÅ : {t R t t 0} denotes the set of real numbers larger or equal to t0. t 0 Æ 2 j ¸ The motivation behind this sequential approach is that LTV systems on the one hand show certain similarities to linear time–invariant systems but are distinguished by significant differences in their dynamic analysis. In this sense, they — at least to some extend — resemble nonlinear systems. On the other hand, the linearization of a nonlinear system around a solution trajectory t (x¤(t),u¤(t)) 7! yields an LTV system. Hence the subsequently introduced tools can be used for control and observer design at least in a neighborhood of the trajectory (x¤(t),u¤(t)). 2.1 Transition matrix and solution of the state differential equations At first free or autonomous systems x˙ A(t)x, t t , x(t ) x (2.2) Æ È 0 0 Æ 0 are addressed.
    [Show full text]
  • Learning Stable Dynamical Systems Using Contraction Theory
    Learning Stable Dynamical Systems using Contraction Theory eingereichte MASTERARBEIT von cand. ing. Caroline Blocher geb. am 19.08.1990 wohnhaft in: Leonrodstrasse 72 80636 M¨unchen Tel.: 0176 27250499 Lehrstuhl f¨ur STEUERUNGS- und REGELUNGSTECHNIK Technische Universit¨atM¨unchen Univ.-Prof. Dr.-Ing./Univ. Tokio Martin Buss Fachgebiet f¨ur DYNAMISCHE MENSCH-ROBOTER-INTERAKTION f¨ur AUTOMATISIERUNGSTECHNIK Technische Universit¨atM¨unchen Prof. Dongheui Lee, Ph.D. Betreuer: M.Sc. Matteo Saveriano Beginn: 03.08.2015 Zwischenbericht: 25.09.2015 Abgabe: 19.01.2016 In your final hardback copy, replace this page with the signed exercise sheet. Abstract This report discusses the learning of robot motion via non-linear dynamical systems and Gaussian Mixture Models while optimizing the trade-off between global stability and accurate reproduction. Contrary to related work, the approach used in this thesis seeks to guarantee the stability via Contraction Theory. This point of view allows the use of results in robust control theory and switched linear systems for the analysis of the global stability of the dynamical system. Furthermore, a modification of existing approaches to learn a globally stable system and an approach to locally stabilize an already learned system are proposed. Both approaches are based on Contraction Theory and are compared to existing methods. Zusammenfassung Diese Arbeit behandelt das Lernen von stabilen dynamischen Systemen ¨uber eine Gauss’sche Mischverteilung. Im Gegensatz zu bisherigen Arbeiten wird die Sta- bilit¨at des Systems mit Hilfe der Contraction Theory untersucht. Ergebnisse aus der robusten Regulung und der Stabilit¨at von schaltenden Systemen k¨onnen so ¨ubernommen werden. Um die Stabilit¨at des dynamischen Systems zu garantieren und gleichzeitig die Bewegung des gelernten Systems m¨oglichst wenig zu beein- flussen, wird eine Anpassung der bereits bestehenden Methode an die Bewegung vorgeschlagen.
    [Show full text]
  • Linear Operators Hsiu-Hau Lin [email protected] (Mar 25, 2010)
    Linear Operators Hsiu-Hau Lin [email protected] (Mar 25, 2010) The notes cover linear operators and discuss linear independence of func- tions (Boas 3.7-3.8). • Linear operators An operator maps one thing into another. For instance, the ordinary func- tions are operators mapping numbers to numbers. A linear operator satisfies the properties, O(A + B) = O(A) + O(B);O(kA) = kO(A); (1) where k is a number. As we learned before, a matrix maps one vector into another. One also notices that M(r1 + r2) = Mr1 + Mr2;M(kr) = kMr: Thus, matrices are linear operators. • Orthogonal matrix The length of a vector remains invariant under rotations, ! ! x0 x x0 y0 = x y M T M : y0 y The constraint can be elegantly written down as a matrix equation, M T M = MM T = 1: (2) In other words, M T = M −1. For matrices satisfy the above constraint, they are called orthogonal matrices. Note that, for orthogonal matrices, computing inverse is as simple as taking transpose { an extremely helpful property for calculations. From the product theorem for the determinant, we immediately come to the conclusion det M = ±1. In two dimensions, any 2 × 2 orthogonal matrix with determinant 1 corresponds to a rotation, while any 2 × 2 orthogonal HedgeHog's notes (March 24, 2010) 2 matrix with determinant −1 corresponds to a reflection about a line. Let's come back to our good old friend { the rotation matrix, cos θ − sin θ ! cos θ sin θ ! R(θ) = ;RT = : (3) sin θ cos θ − sin θ cos θ It is straightforward to check that RT R = RRT = 1.
    [Show full text]
  • Generalized Hurwitz Matrices, Generalized Euclidean Algorithm
    Generalized Hurwitz matrices, generalized Euclidean algorithm, and forbidden sectors of the complex plane Olga Holtz∗†, Sergey Khrushchev,‡ and Olga Kushel§ June 23, 2015 Abstract Given a polynomial n n−1 f(x)= a0x + a1x + · · · + an with positive coefficients ak, and a positive integer M ≤ n, we define a(n infinite) generalized Hurwitz matrix HM (f):=(aMj−i)i,j . We prove that the polynomial f(z) does not vanish in the sector π z ∈ C : | arg(z)| < n M o whenever the matrix HM is totally nonnegative. This result generalizes the classical Hurwitz’ Theo- rem on stable polynomials (M = 2), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots (M = 1), and the Cowling-Thron theorem (M = n). In this connection, we also develop a generalization of the classical Euclidean algorithm, of independent interest per se. Introduction The problem of determining the number of zeros of a polynomial in a given region of the complex plane is very classical and goes back to Descartes, Gauss, Cauchy [4], Routh [22, 23], Hermite [9], Hurwitz [14], and many others. The entire second volume of the delightful Problems and Theorems in Analysis by P´olya and Szeg˝o[21] is devoted to this and related problems. See also comprehensive monographs of Marden [17], Obreshkoff [19], and Fisk [6]. One particularly famous late-19th-century result, which also has numerous applications, is the Routh- Hurwitz criterion of stability. Recall that a polynomial is called stable if all its zeros lie in the open left half-plane of the complex plane. The Routh-Hurwitz criterion asserts the following: Theorem 1 (Routh-Hurwitz [14, 22, 23]).
    [Show full text]
  • Inclusion Relations Between Orthostochastic Matrices And
    Linear and Multilinear Algebra, 1987, Vol. 21, pp. 253-259 Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 Photocopying permitted by license only 1987 Gordon and Breach Science Publishers, S.A. Printed in the United States of America Inclusion Relations Between Orthostochastic Matrices and Products of Pinchinq-- Matrices YIU-TUNG POON Iowa State University, Ames, lo wa 5007 7 and NAM-KIU TSING CSty Poiytechnic of HWI~K~tig, ;;e,~g ,?my"; 2nd A~~KI:Lkwe~ity, .Auhrn; Al3b3m-l36849 Let C be thc set of ail n x n urthusiuihaatic natiiccs and ;"', the se! of finite prnd~uctpof n x n pinching matrices. Both sets are subsets of a,, the set of all n x n doubly stochastic matrices. We study the inclusion relations between C, and 9'" and in particular we show that.'P,cC,b~t-~p,t~forn>J,andthatC~$~~fori123. 1. INTRODUCTION An n x n matrix is said to be doubly stochastic (d.s.) if it has nonnegative entries and ail row sums and column sums are 1. An n x n d.s. matrix S = (sij)is said to be orthostochastic (0.s.) if there is a unitary matrix U = (uij) such that 2 sij = luijl , i,j = 1, . ,n. Following [6], we write S = IUI2. The set of all n x n d.s. (or 0s.) matrices will be denoted by R, (or (I,, respectively). If P E R, can be expressed as 1-t (1 2 54 Y. T POON AND N K. TSING Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 for some permutation matrix Q and 0 < t < I, then we call P a pinching matrix.
    [Show full text]